Let `a ,b ,c in Q^+` satisfying `a > b > cdot` Which of the following statements (s) hold true of the quadratic polynomial `f(x)=(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)?` The mouth of the parabola `y=f(x)` opens upwards Both roots of the equation `f(x)=0` are rational The x-coordinate of vertex of the graph is positive The product of the roots is always negative

Options

A. The mouth of the parabola f(x) = 0 opens upwards
B. Both roots of the equation f(x) = 0 are rational
C. The x-coordinate of vertex of the graph is positive
D. The product of the roots is always negative .

Answer

1,2,3,

Explanation

` f(x) = Ax^(2) + Bx + C`
` A = a + b - 2c = (a -c) + (b - c) gt 0 `
`rArr A gt 0 `
Hence, the graph is concave upwards. Also, x = 1 is obvious
Solution, therefore .both roots are rational
` b +c - 2a = underset(-ve)underbrace(b - a)+ underset(-ve)underbrace(c - a)lt 0 `
`rArr B lt 0 `
` therefore - (B)/(2A)gt 0 `
Hence , abscissa of the vertex is positive . Option (d) need not be
correct as with a = 5 ,b = 4 , c = 2 , P` lt ` 0 and with a = 6 , b = 3
c = 2, P ` gt ` 0 .